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Claude Fable produced a counterexample to the Jacobian Conjecture

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Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#331

Post by regularfry »

dist-epoch wrote: post #323
raxxorraxor wrote: post #309

Earlier quoted context omitted.

Those silly advertisers do everything for exposure and if that means digging yourself into a niche alleged mathematical theorem to refute it, it is what needs to be done! Of course it would be really interesting how Claude approached this. Probably with some constraints regarding the input. And it would be interesting what these constraints were.

Author probably doesn't want to show the prompt because they are now trying to find a bunch of other counter-examples with the same prompt

The reasoning trace would be far more interesting, and that's not exposed.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#332

Post by pred_ »

glimshe wrote: post #264
antirez wrote: post #254

Earlier quoted context omitted.

So many mathematicians over the years tried hard and failed, but now Anthropic just for some PR magically did it? And this after LLMs obtaining different math wins? What is your logic here really escapes my understanding.

The parent's absolutely nonsensical post highlights how polarized AI (as everything else) is today. I can understand someone being opposed to AI on moral, cost-benefit or productivity grounds. But we're seeing a lot of extremist "AI is good for nothing" posts out there nowadays.

I don't think it is nonsensical at all. The author and his collaborator both appear to be bright people, so there's a good chance they had to offer non-trivial insights to guide the LLM, yet it's clearly in the interest of his employer to downplay whatever personal contribution they provided.

Edit: Now the OP is flagged/dead for some reason. You could disagree on their take (calling it a marketing stunt is maybe a bit much), but I think the argument is sound, so flagging seems counterproductive to the discussion.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#333

Post by igravious »

William_BB wrote: post #318
eru wrote: post #287

Earlier quoted context omitted.

I'm not so sure. People had been trying to use brute force to find counterexamples before.

AI is great at reducing the search space and using human-like reasoning (in a brute-force way) to carry out the brute-force search. I'm not surprised by this result. This is exactly what AI should excel at, with human guidance.

"using human-like reasoning (in a brute-force way)"

that's self-contradictory -- what brute force means is doing an exhaustive search of a search space (brute forcing it)

using human-like(?) reasoning means cutting down the search space by having some sort of insight or intuition which allows you to prune branches from the entire tree

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#334

Post by dist-epoch »

Lockal wrote: post #316
aizk wrote: post #112

This is a rare instance where feeding this groundbreaking information into an LLM gives _them_ psychosis. I fed this to claude code and watched it verify the result in 7 different ways to be 100% certain, and it was just flabbergasted. Quite remarkable.

I fed it to Google AI Studio, enabling tool execution and disabling web access. It also quickly verified it with SymPy, then went into psychosis. 5 minutes later: all previous chats are loading fine, but the only "Counterexample to the Jacobian Conjecture" chat is not loading. Well, I'm not a conventional conspiracy theorist. But everyone knows that in every major LLM provider there are hell of hidden guarding system…

> all previous chats are loading fine, but the only "Counterexample to the Jacobian Conjecture" chat is not loading.

This happened multiple times to me with Gemini. For the most trivial of requests, like translating a video into English.

> "ground-breaking results that could be attributed to Gemini, if act quick"

This would be such a dumb thing to do, and so easy to get caught with...

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#335

Post by unexpectedtrap »

Using LLMs to generate piles of code and/or proofs of dubious quality is very questionable thing, and I understand these non-stop debates about it.

But in this case, as using plain brute force is already quite a common thing in searching for counterexamples, using LLMs as a sort of more advanced brute force seems to be just the right thing to do, so I struggle to understand so much hostility to this approach.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#337

Post by betafj »

QuesnayJr wrote: post #289
meowface wrote: post #194

Earlier quoted context omitted.

You can use that retroactive logic about any hard problem though. Unsolved murder cases, math, theoretical physics. If tons of smart humans try for years and fail and then an LLM tries for a few weeks or hours and succeeds, the implications are clear. And these are by far the dumbest LLMs will ever be.

The retrospective view is important, though. In retrospect, these problems weren't that hard. (The unit distance graph problem was the hardest.) There are some problems that still seem hard, even when we know the answer. Nobody thinks that Fermat's last theorem is easy, even though now know it to be true. Before AI, it was pretty rare that a problem that turned to be unexpectedly easy, so mathematicians thought they…

If some LLM is able to come up with a simple proof of Fermat's Last Theorem (a solution that Fermat himself could come up with) in the future, would you still say that Fermat's Last Theorem is hard?

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#338

Post by igravious »

ibarrajo wrote: post #203

I’ve been math-vibe coding a few months now. It’s surprisingly easy to do with AI. The hard part has been manually verifying and validating the results. I took one of the smaller findings (disproving a conjecture) and wrote a paper as my first endeavor into publishing. Because the next few findings i have in the pipeline are substantial in the field of quantum topology and physics im taking some time to publish them…

i'm doing the same thing but for type theory

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#339

Post by narrator »

I've been asking Fable very complicated questions in heterodox economic theory, which is something I know a lot about. The stuff it comes back with is incredibly deep.

To give a metaphor that everyone here on HN would understand, reading it's responses gives me the same level of wonder as one gets learning how quicksort works for the first time. It even stretches my brain to grasp what it's even come up with. I find myself getting mentally exhausted just digesting it's brilliance.

I think the singularity will have this point where AI comes up with ideas so profound, like a Ramanujen equation, that the most brilliant among us can't even decipher the answer to our questions. The internal reasoning of the machine is at a level of complexity that's beyond human comprehension to even keep track of everything enough to integrate the understanding of what it's come up with. This will happen with any even mildly complex question about any topic.

Re: Claude Fable produced a counterexample to the Jacobian Conjecture

#340

Post by tptacek »

aleph_minus_one wrote: post #312
tptacek wrote: post #61

Earlier quoted context omitted.

I'm pretty sure you can just check this for yourself. The Sage POC is like 8 lines.

This assumes that you already have Sage installed, and you have some working knowledge in using Sage.

You also have to know how to take a Jacobian, so I can see how this might be a live controversy on HN.
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